Suppose we are in the setting of Theorem REF , so \(M\)  is a \(C^r\)  manifold of dimesion \(2(m+n)\) , and \(F \in \mathrm {Diff}^r (M)\)  has a normally hyperbolic invariant manifold \(\Lambda \simeq \mathbb {T}^n \times [0,1]^n\)  in \(M\)  satisfying [A1-3]. In order to construct correctly aligned windows, we need estimates for the map in a neighbourhood of the invariant manifolds \(W^{s,u}(\Lambda )\) . This requires an appropriate system of coordinates in which the twist property of the inner map \(F|_{\Lambda }\)  is apparent; Fenichel coordinates (described below; see [1]}) provide a starting point for the coordinate transformation. However, when we express the map \(F\)  in Fenichel coordinates, there are error terms that complicate the estimates. We therefore seek a further coordinate transformation in which these error terms can be ignored. There is an analogue of the Hartman-Grobman Theorem for normally hyperbolic invariant manifolds, which says that there is a neighbourhood of \(\Lambda \)  in which \(F\)  is topologically conjugate to its so-called normal map [2]}. These coordinates would be ideal for obtaining local estimates in a neighbourhood of the cylinder (as in Section REF ), but later we need to analyse how the images of this coordinate chart fit together at a homoclinic point if we iterate it backwards (resp. forwards) along the stable (resp. unstable) manifold of \(\Lambda \)  (see Section REF ). This analysis requires a continuous second derivative; since the coordinates provided by [2]} are purely topological, we instead follow Section 5.1 of [4]} to construct a suitable system of coordinates that are as smooth as required.
